Worked example · Easy
Temperature change
The temperature rises from -4°C to 7°C. Find the increase.
\[7-(-4)=11\]
Increase: \(\boxed{11^\circ\text{C}}\).
Chapter 6
The four operations
Build reliable arithmetic with negative numbers, integers, decimals and fractions. The aim is not only to calculate correctly, but to choose the right operation, apply the correct sign and follow the correct order.
By the end of this chapter, you should be able to
- interpret negative values in real contexts and calculate differences across zero;
- add, subtract, multiply and divide positive and negative numbers;
- apply brackets, powers, multiplication/division and addition/subtraction in the correct order;
- multiply and divide large integers using reliable written methods;
- multiply and divide decimals while placing the decimal point correctly;
- add and subtract fractions using common denominators;
- multiply fractions efficiently by cancelling common factors;
- divide fractions by multiplying by the reciprocal;
- simplify final fraction answers and convert improper fractions to mixed numbers when appropriate.
1
Subtopic 1
Numbers below zero
Negative numbers are numbers less than zero. They appear in temperatures, bank balances, heights below sea level, lifts below ground level and many other contexts. Zero is neither positive nor negative.
-5-4-3-2-1012345
Difference between two values
If the values are on opposite sides of zero, count the distance to zero and then beyond it. Algebraically, the difference between 8 and -5 is:
\[8-(-5)=13.\]Exam method
Distances and changes across zero must include both sides of zero.
Worked example · Exam-style
Bank balance across zero
An account is $68 overdrawn. How much must be paid in for the balance to become $125 in credit?
Start at \(-68\), finish at \(+125\).
\[125-(-68)=193\]Amount required: \(\boxed{\$193}\).
Worked example · Exam-style extension
Find a change across zero
A submarine moves from \(-85\text{ m}\) to \(27\text{ m}\). Find the change in its height.
Change = final height minus starting height.
\[27-(-85)=27+85=\boxed{112\text{ m upward}}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Which is colder: \(-12^\circ\text{C}\) or \(-7^\circ\text{C}\)?
Show Answer
On the number line, \(-12<-7\).
\[\boxed{-12^\circ\text{C}}\]
2Apply the method[4 marks]
Calculate the temperature rise from \(-6^\circ\text{C}\) to \(9^\circ\text{C}\).
Show Answer
\[9-(-6)=9+6=\boxed{15^\circ\text{C}}\]
3Exam-style challenge[5 marks]
An account goes from \(-\$45\) to \(\$120\). How much has the balance increased?
Show Answer
\[120-(-45)=120+45=\boxed{\$165}\]
2
Subtopic 2
Adding and subtracting with negative numbers
A number line helps explain the rules. Adding a negative number moves left; subtracting a negative number has the same effect as adding the corresponding positive number.
Adding a negative
\(a+(-b)=a-b\)
Subtracting a negative
\(a-(-b)=a+b\)
Exam method
Keep track of the operation separately from the sign attached to a number.
Worked example · Easy
Add a negative number
Work out \(-3+(-7)\).
\[-3+(-7)=-3-7=\boxed{-10}\]
Worked example · Medium
Subtract a negative number
Work out \(-6-(-9)\).
\[-6-(-9)=-6+9=\boxed{3}\]
Worked example · Exam-style extension
Track several signed changes
Evaluate \(-15+(-8)-(-11)+6\).
Change addition of a negative to subtraction; change subtraction of a negative to addition.
\[-15-8+11+6=-23+17=\boxed{-6}\]
Common mistake
The rule “two negatives make a positive” is not a general addition rule. It applies here because you are subtracting a negative, and later to multiplication/division of two negative factors.
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Calculate \(-9+(-4)\).
Show Answer
Add two negative values by adding their magnitudes.
\[-9-4=\boxed{-13}\]
2Apply the method[4 marks]
Calculate \(7-(-12)-5\).
Show Answer
\[7+12-5=19-5=\boxed{14}\]
3Exam-style challenge[5 marks]
A lift is on floor \(-3\), goes up 8 floors, then down 11 floors. Which floor?
Show Answer
\[-3+8-11=5-11=\boxed{-6}\]
3
Subtopic 3
Multiplying and dividing with negative numbers
For multiplication and division, the sign depends only on whether the signs are the same or different.
| Signs | Result |
|---|---|
| same: \(+\times+\), \(-\times-\), \(+\div+\), \(-\div-\) | positive |
| different: \(+\times-\), \(-\times+\), \(+\div-\), \(-\div+\) | negative |
Several negative factors
An even number of negative factors gives a positive product. An odd number gives a negative product.
Exam method
An odd number of negative factors gives a negative product; an even number gives a positive product.
Worked example · Easy
Two negative factors
Work out \((-8)(-6)\).
Same signs → positive.
\[8\times6=48\Rightarrow\boxed{48}\]Worked example · Medium
Three factors
Work out \((-2)\times5\times(-7)\).
There are two negative signs, so the result is positive.
\[2\times5\times7=\boxed{70}\]Worked example · Exam-style extension
Multiply a sequence of signed values
Evaluate \((-2)(-3)(-4)(5)\).
There are three negative factors, so the final product is negative.
\[2\times3\times4\times5=120\]
\[\boxed{-120}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Calculate \((-7)\times6\).
Show Answer
The signs differ, so the product is negative.
\[\boxed{-42}\]
2Apply the method[4 marks]
Calculate \((-144)\div(-12)\).
Show Answer
Two negative signs give a positive quotient.
\[\boxed{12}\]
3Exam-style challenge[5 marks]
Evaluate \((-2)(-5)(-3)(-4)\).
Show Answer
Four negative factors give a positive product.
\[2\times5\times3\times4=\boxed{120}\]
4
Subtopic 4
Combining operations
When a calculation contains several operations, multiplication and division are completed before addition and subtraction unless brackets change the order.
Exam method
In a mixed operation calculation, resolve multiplication and division before addition and subtraction.
Worked example · Medium
Mixed signs and operations
Work out \((-4\times-3)+(5\times-2)\).
\[(-4\times-3)=12,\qquad5\times-2=-10\]\[12+(-10)=\boxed{2}\]
Worked example · Exam-style
Numerator and denominator
Work out \(\frac{6(-5)+4(-3)}{-7+1}\).
Numerator:
\[6(-5)+4(-3)=-30-12=-42\]Denominator:
\[-7+1=-6\]Then divide:
\[\frac{-42}{-6}=\boxed{7}\]Worked example · Exam-style extension
Combine operations carefully
Work out \(18-(-3)(4)+(-24)\div6\).
Do multiplication and division first: \((-3)(4)=-12\), \((-24)\div6=-4\).
\[18-(-12)+(-4)=18+12-4=\boxed{26}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Evaluate \(12+(-3)\times4\).
Show Answer
Multiply first: \((-3)\times4=-12\).
\[12-12=\boxed0\]
2Apply the method[4 marks]
Calculate \(7-(-18)\div3\).
Show Answer
Division first: \((-18)\div3=-6\).
\[7-(-6)=\boxed{13}\]
3Exam-style challenge[5 marks]
Evaluate \((-24)\div(-6)+3(-5)-(-2)\).
Show Answer
Divide and multiply: \(4+(-15)-(-2)\).
\[4-15+2=\boxed{-9}\]
5
Subtopic 5
Order of operations
Use the same order every time. A useful memory aid is BIDMAS (or BODMAS/PEMDAS in other systems).
1Brackets
2Indices / powers
3Division and multiplication
4Addition and subtraction
Operations at the same level are done from left to right. A fraction bar acts like brackets around the entire numerator and denominator.
Exam method
Brackets first; powers next; multiply and divide left to right; add and subtract left to right.
Worked example · Easy
Multiplication before addition
Work out \(9-2\times4\).
\[9-2\times4=9-8=\boxed{1}\]
Worked example · Exam-style
Brackets, power and a negative product
Work out \((7-3)^2+4\times(-5)\).
\[(7-3)^2+4(-5)=4^2-20=16-20=\boxed{-4}\]
Worked example · Exam-style extension
Order and equal-precedence rules
Evaluate \(36\div3\times2-(5-2)^2\).
Brackets and indices first: \((5-2)^2=3^2=9\).
Division and multiplication have equal priority: go left to right.
\[36\div3\times2=12\times2=24\]
\[24-9=\boxed{15}\]
Calculator warning
Enter brackets exactly as written. Do not rely on a calculator to repair missing brackets or an incorrectly typed negative sign.
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Calculate \(8+6\times3\).
Show Answer
Multiply before adding: \(6\times3=18\).
\[8+18=\boxed{26}\]
2Apply the method[4 marks]
Evaluate \((9-5)^2+3\times2\).
Show Answer
\[(9-5)^2=16,\quad3\times2=6\]
\[16+6=\boxed{22}\]
3Exam-style challenge[5 marks]
Calculate \(40\div(6-2)\times3-2^3\).
Show Answer
Bracket \(6-2=4\), power \(2^3=8\).
\[40\div4\times3-8=10\times3-8=\boxed{22}\]
6
Subtopic 6
Multiplying integers
For large integers, use a written method such as long multiplication or a grid. Both rely on place value and should give the same result.
Exam method
Estimate products first and align place values correctly in written multiplication.
Worked example · Medium
Long multiplication by place value
Work out \(426\times37\).
Split 37 into 30 and 7:
\[426\times30=12780\]\[426\times7=2982\]\[12780+2982=\boxed{15762}\]Worked example · Exam-style
Use multiplication in context
A hall has 48 rows with 36 seats in each row. Find the total number of seats.
\[48\times36=48(30+6)=1440+288=\boxed{1728}\]
Worked example · Exam-style extension
Place-value multiplication of integers
Calculate \(608\times27\) using place value.
Split \(27=20+7\).
\[608\times20=12160\]
\[608\times7=4256\]
\[12160+4256=\boxed{16416}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Work out \(35\times24\).
Show Answer
\[35(20+4)=700+140=\boxed{840}\]
2Apply the method[4 marks]
Find \(214\times36\).
Show Answer
\[214(30+6)=6420+1284=\boxed{7704}\]
3Exam-style challenge[5 marks]
A warehouse stores 48 cartons, each holding 125 notebooks. How many notebooks altogether?
Show Answer
\[48\times125=(50-2)125=6250-250=\boxed{6000}\]
7
Subtopic 7
Multiplying decimals
Ignore the decimal points while multiplying, then place the decimal point in the answer so that the total number of decimal places matches the total in the two factors.
Reliable method
- Count the decimal places in both factors altogether.
- Multiply as if the numbers were integers.
- Put the decimal point into the product so the answer has the required number of decimal places.
- Estimate to check whether the size of the answer is sensible.
Exam method
Estimate the size of your product: multiplying by a decimal smaller than 1 decreases a positive quantity.
Worked example · Easy
Small decimal product
Work out \(0.6\times0.04\).
There are three decimal places altogether.
\[6\times4=24\Rightarrow0.024\]\(\boxed{0.024}\)
Worked example · Medium
Two decimal factors
Work out \(3.24\times0.7\).
\[324\times7=2268\]
There are three decimal places altogether:
\[\boxed{2.268}\]Worked example · Exam-style extension
Decimal placement and estimation
Work out \(3.25\times0.48\).
First multiply integers: \(325\times48=15600\).
There are 2 + 2 = 4 decimal places in the factors.
\[3.25\times0.48=\boxed{1.56}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Calculate \(0.7\times0.06\).
Show Answer
\(7\times6=42\) and there are 3 decimal places.
\[\boxed{0.042}\]
2Apply the method[4 marks]
Evaluate \(2.4\times1.25\).
Show Answer
\[24\times125=3000\] Three decimal places in total.
\[\boxed{3}\]
3Exam-style challenge[5 marks]
Paint covers \(6.25\text{ m}^2\) per litre. How much area can \(3.6\) litres cover?
Show Answer
\[6.25\times3.6=6.25(3+0.6)=18.75+3.75\]
\[\boxed{22.5\text{ m}^2}\]
8
Subtopic 8
Dividing integers
Long division or chunking can be used. For a context question, always interpret the remainder: sometimes it means another container, bus or box is needed.
Exam method
An interpretation question may require rounding the number of buses or boxes up, rather than reporting a remainder.
Worked example · Medium
Exact integer division
Work out \(936\div26\).
\(26\times30=780\), leaving 156.
\(26\times6=156\).
\[936\div26=30+6=\boxed{36}\]Worked example · Exam-style
Interpret a remainder
625 students travel in coaches that each hold 48 students. What is the minimum number of coaches needed?
\[625\div48=13\text{ remainder }1\]
13 coaches are not enough because one student remains.
Minimum needed: \(\boxed{14}\).
Worked example · Exam-style extension
A division result with remainder
How many full groups of 28 can be made from 997, and how many are left?
\(28\times35=980\).
Remainder \(997-980=17\).
\[\boxed{35\text{ groups, remainder }17}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Compute \(672\div21\).
Show Answer
\(21\times32=672\).
\[\boxed{32}\]
2Apply the method[4 marks]
Calculate \(935\div24\), stating quotient and remainder.
Show Answer
\(24\times38=912\), leaving \(935-912=23\).
\[\boxed{38\text{ remainder }23}\]
3Exam-style challenge[5 marks]
There are 1245 people and each minibus seats 32. Find the minimum number of minibuses needed.
Show Answer
\(32\times38=1216\), leaving 29 people.
They need another minibus: \(38+1=\boxed{39}\).
9
Subtopic 9
Dividing decimals
Make the divisor an integer by multiplying both numbers by the same power of 10. This does not change the value of the division.
Exam method
Always move the decimal point the same number of places in BOTH numbers.
Worked example · Easy
Decimal divisor
Work out \(4.8\div0.6\).
\[4.8\div0.6=48\div6=\boxed{8}\]
Worked example · Exam-style
Two decimal places in the divisor
Work out \(18.72\div0.24\).
Multiply both numbers by 100:
\[18.72\div0.24=1872\div24=\boxed{78}\]Worked example · Exam-style extension
Divide both values by the same scale
Calculate \(0.0468\div0.012\).
Multiply the divisor and dividend by 1000.
\[0.0468\div0.012=46.8\div12=\boxed{3.9}\]
Common mistake
If you move the decimal point in the divisor, you must move it by the same number of places in the dividend.
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Calculate \(7.2\div0.9\).
Show Answer
Multiply both numbers by 10: \(72\div9\).
\[\boxed8\]
2Apply the method[4 marks]
Work out \(4.86\div0.18\).
Show Answer
Multiply both values by 100: \(486\div18\).
\[\boxed{27}\]
3Exam-style challenge[5 marks]
\(5.4\) litres of juice are poured into cups holding \(0.15\) litres each. How many complete cups?
Show Answer
\[5.4\div0.15=540\div15=\boxed{36\text{ cups}}\]
10
Subtopic 10
Adding and subtracting fractions
Fractions must have a common denominator before their numerators can be added or subtracted. Mixed numbers are safest to convert to improper fractions first.
Method
- Convert mixed numbers to improper fractions if necessary.
- Find a common denominator (preferably the LCM).
- Write equivalent fractions.
- Add or subtract the numerators.
- Simplify and convert an improper final answer to a mixed number if appropriate.
Exam method
Find a common denominator first, calculate with numerators, then simplify the answer.
Worked example · Easy
Add unlike fractions
Work out \(\frac34+\frac56\).
LCM of 4 and 6 is 12.
\[\frac34+\frac56=\frac9{12}+\frac{10}{12}=\frac{19}{12}=\boxed{1\frac7{12}}\]Worked example · Exam-style
Subtract mixed numbers
Work out \(4\frac25-1\frac34\).
\[4\frac25=\frac{22}{5},\qquad1\frac34=\frac74\]\[\frac{22}{5}-\frac74=\frac{88}{20}-\frac{35}{20}=\frac{53}{20}=\boxed{2\frac{13}{20}}\]
Worked example · Exam-style extension
Mixed fractions with unlike denominators
Calculate \(2\frac34-1\frac56+\frac23\).
Use improper fractions: \(\frac{11}{4}-\frac{11}{6}+\frac23\).
Common denominator 12:
\[\frac{33}{12}-\frac{22}{12}+\frac8{12}=\frac{19}{12}\]
\[\boxed{1\frac7{12}}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Find \(\frac13+\frac14\).
Show Answer
Common denominator 12.
\[\frac4{12}+\frac3{12}=\boxed{\frac7{12}}\]
2Apply the method[4 marks]
Work out \(\frac78-\frac13\).
Show Answer
Common denominator 24.
\[\frac{21}{24}-\frac8{24}=\boxed{\frac{13}{24}}\]
3Exam-style challenge[5 marks]
A jug contains \(2\frac12\) litres. \(\frac34\) litre and \(\frac56\) litre are used. How much remains?
Show Answer
\[\frac52-\frac34-\frac56=\frac{30-9-10}{12}\]
\[\boxed{\frac{11}{12}\text{ litre}}\]
11
Subtopic 11
Multiplying fractions
Multiply the numerators and multiply the denominators. Cancelling common factors before multiplying keeps the numbers smaller.
Exam method
Simplifying factors before multiplying is useful when the numerators and denominators are large.
Worked example · Easy
Fraction by a whole number
Work out \(\frac7{15}\times6\).
\[\frac7{15}\times\frac61=\frac{7\times2}{5}=\boxed{\frac{14}{5}=2\frac45}\]
Worked example · Medium
Cancel before multiplying
Work out \(\frac{9}{14}\times\frac{21}{20}\).
Cancel \(21\div14=3\div2\) and \(9\div20\) has no common factor.
\[\frac9{14}\times\frac{21}{20}=\frac{9\times3}{2\times20}=\frac{27}{40}\]\(\boxed{\frac{27}{40}}\)
Worked example · Exam-style
Mixed-number multiplication
Work out \(2\frac14\times1\frac23\).
\[2\frac14=\frac94,\qquad1\frac23=\frac53\]\[\frac94\times\frac53=\frac{45}{12}=\frac{15}{4}=\boxed{3\frac34}\]
Worked example · Exam-style extension
Cancel factors before multiplying
Calculate \(\frac{15}{28}\times\frac{14}{25}\).
Cancel 14 with 28 to get \(1\) and \(2\); cancel 15 with 25 to get \(3\) and \(5\).
\[\frac{15}{28}\times\frac{14}{25}=\frac{3\times1}{2\times5}=\boxed{\frac3{10}}\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Evaluate \(\frac34\times\frac25\).
Show Answer
Multiply numerators and denominators.
\[\frac{3\times2}{4\times5}=\boxed{\frac3{10}}\]
2Apply the method[4 marks]
Find \(1\frac12\times\frac49\).
Show Answer
Change the mixed number: \(1\frac12=\frac32\).
\[\frac32\times\frac49=\frac{12}{18}=\boxed{\frac23}\]
3Exam-style challenge[5 marks]
A rectangular field is \(\frac78\text{ km}\) long and \(\frac25\text{ km}\) wide. Find its area.
Show Answer
\[\frac78\times\frac25=\frac{14}{40}=\boxed{\frac7{20}\text{ km}^2}\]
12
Subtopic 12
Dividing fractions
To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac ab\) is \(\frac ba\).
Keep – change – flip
Keep the first fraction, change division to multiplication, flip the second fraction. This memory aid is useful, but you should understand that “flip” means taking the reciprocal.
Exam method
Only invert the divisor (the second fraction); convert mixed numbers to improper fractions first.
Worked example · Easy
Divide by a unit fraction
Work out \(5\div\frac14\).
\[5\div\frac14=5\times4=\boxed{20}\]
Worked example · Medium
Fraction divided by fraction
Work out \(\frac56\div\frac7{12}\).
\[\frac56\times\frac{12}{7}=\frac{5\times2}{7}=\boxed{\frac{10}{7}=1\frac37}\]
Worked example · Exam-style
Mixed-number division
Work out \(3\frac38\div1\frac12\).
\[3\frac38=\frac{27}{8},\qquad1\frac12=\frac32\]\[\frac{27}{8}\div\frac32=\frac{27}{8}\times\frac23=\frac{54}{24}=\frac94=\boxed{2\frac14}\]
Worked example · Exam-style extension
Divide a mixed number by a fraction
Calculate \(2\frac14\div\frac38\).
Convert the mixed number: \(2\frac14=\frac94\).
Multiply by the reciprocal \(\frac83\).
\[\frac94\times\frac83=\frac{72}{12}=\boxed6\]
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Calculate \(3\div\frac12\).
Show Answer
\[3\times\frac21=\boxed6\]
2Apply the method[4 marks]
Work out \(\frac59\div\frac{10}{27}\).
Show Answer
Multiply by the reciprocal and cancel.
\[\frac59\times\frac{27}{10}=\frac{15}{10}=\boxed{\frac32}\]
3Exam-style challenge[5 marks]
A ribbon \(2\frac25\text{ m}\) long is cut into \(\frac3{10}\text{ m}\) pieces. How many complete pieces?
Show Answer
\[\frac{12}5\div\frac3{10}=\frac{12}5\times\frac{10}3=\boxed8\]
Chapter 6 practice
The first questions build fluency. The later questions combine several skills, as exam questions often do. Keep each answer closed until you have attempted the question.
A. Negative numbers and signs
1Temperature difference[2 marks]
Find the difference between \(-12^\circ\text{C}\) and \(5^\circ\text{C}\).
Show Answer
\[5-(-12)=\boxed{17^\circ\text{C}}\]
2Add and subtract negatives[4 marks]
Work out: a) \(-7+(-8)\) b) \(3-(-11)\) c) \(-4-6\) d) \(-9-(-2)\).
Show Answer
a) \(-15\)
b) \(14\)
c) \(-10\)
d) \(-7\)
3Multiply and divide signs[4 marks]
Work out: a) \((-6)(7)\) b) \((-8)(-5)\) c) \(54\div(-9)\) d) \((-72)\div(-8)\).
Show Answer
a) \(-42\)
b) \(40\)
c) \(-6\)
d) \(9\)
B. Order and written arithmetic
4Order of operations[3 marks]
Work out \(18-3\times4+2\).
Show Answer
Multiplication first:
\[18-12+2=\boxed{8}\]5Brackets and powers[4 marks]
Work out \((9-5)^2-3\times6\).
Show Answer
\[(9-5)^2-18=4^2-18=16-18=\boxed{-2}\]
6Integer multiplication[3 marks]
Work out \(538\times46\).
Show Answer
\[538\times40=21520\]\[538\times6=3228\]\[21520+3228=\boxed{24748}\]
7Integer division[3 marks]
Work out \(1152\div32\).
Show Answer
\(32\times30=960\), remainder 192. \(32\times6=192\).
\[1152\div32=\boxed{36}\]C. Decimal operations
8Multiply decimals[3 marks]
Work out \(0.35\times0.8\).
Show Answer
\[35\times8=280\]
Three decimal places altogether:
\[\boxed{0.280=0.28}\]9Divide decimals[3 marks]
Work out \(7.56\div0.18\).
Show Answer
\[7.56\div0.18=756\div18=\boxed{42}\]
10Mixed decimal expression[4 marks]
Work out \(0.7+0.4\times0.25\).
Show Answer
Multiply first:
\[0.4\times0.25=0.1\]\[0.7+0.1=\boxed{0.8}\]D. Fraction operations
11Add fractions[3 marks]
Work out \(\frac5{12}+\frac7{18}\).
Show Answer
LCM of 12 and 18 is 36.
\[\frac{15}{36}+\frac{14}{36}=\boxed{\frac{29}{36}}\]12Subtract mixed numbers[4 marks]
Work out \(5\frac16-2\frac34\).
Show Answer
\[5\frac16=\frac{31}{6},\quad2\frac34=\frac{11}{4}\]\[\frac{62}{12}-\frac{33}{12}=\frac{29}{12}=\boxed{2\frac5{12}}\]
13Multiply fractions[3 marks]
Work out \(\frac8{15}\times\frac9{16}\).
Show Answer
Cancel \(8\) with \(16\), and 9 with 15:
\[\frac12\times\frac35=\boxed{\frac3{10}}\]14Divide fractions[3 marks]
Work out \(\frac7{10}\div\frac{14}{15}\).
Show Answer
\[\frac7{10}\times\frac{15}{14}=\frac12\times\frac32=\boxed{\frac34}\]
15Mixed-number multiplication[4 marks]
Work out \(1\frac45\times2\frac7{9}\).
Show Answer
\[1\frac45=\frac95,\quad2\frac79=\frac{25}{9}\]\[\frac95\times\frac{25}{9}=\frac{25}{5}=\boxed{5}\]
E. Exam-style challenge
16Bank account[5 marks]
A bank balance is \(-\$145\). A payment of $230 is deposited and then $96 is withdrawn. Find the final balance.
Show Answer
\[-145+230=85\]\[85-96=\boxed{-\$11}\]
17Coach problem[4 marks]
738 people must travel in coaches holding 52 people each. Find the minimum number of coaches required.
Show Answer
\[738\div52=14\text{ remainder }10\]
One more coach is needed.
\(\boxed{15}\) coaches.
18Combined operations[5 marks]
Work out \(\frac{18-3^2}{5-8}\).
Show Answer
Numerator:
\[18-3^2=18-9=9\]Denominator:
\[5-8=-3\]\[\frac9{-3}=\boxed{-3}\]19Fraction expression[6 marks]
Work out \(2\frac12+\frac34\div\frac98\). Give the answer in simplest form.
Show Answer
Do the division first:
\[\frac34\div\frac98=\frac34\times\frac89=\frac23\]Then add:
\[2\frac12+\frac23=\frac52+\frac23=\frac{15}{6}+\frac46=\frac{19}{6}=\boxed{3\frac16}\]20Decimal and order of operations[5 marks]
Work out \(2.4-0.6\div0.15\times0.3\).
Show Answer
Division and multiplication are at the same level, so work left to right:
\[0.6\div0.15=4\]\[4\times0.3=1.2\]\[2.4-1.2=\boxed{1.2}\]21Error spotting[4 marks]
A student writes \(6+2\times5=40\). Explain the error and give the correct answer.
Show Answer
The student added first, but multiplication must be done before addition.
\[6+2\times5=6+10=\boxed{16}\]22Fraction context[5 marks]
A container is \(\frac34\) full. Then \(\frac25\) of the liquid currently in the container is used. What fraction of the full container remains?
Show Answer
Using \(\frac25\) means \(\frac35\) remains.
\[\frac35\times\frac34=\frac9{20}\]\(\boxed{\frac9{20}}\) of the full container remains.
Chapter 6 high-grade checklist
Negative values
I can move across zero and calculate differences in context.
Signs
I distinguish addition/subtraction rules from multiplication/division sign rules.
Order
I apply brackets, powers, multiplication/division, then addition/subtraction.
Integers
I can multiply and divide large integers using a written method and interpret remainders.
Decimals
I place decimal points using place value and make a decimal divisor into an integer.
Fractions
I use common denominators for add/subtract, cancel for multiply, and reciprocals for divide.
Illustrations are taken only from actual Chapter 6 diagrams where present; explanations and practice questions are newly written. Text-only book pages are not reproduced as images.